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\centerline{\Large Math 675 Homework 8}
\centerline{Due 10/31/2018}
\vspace{.25in}

\begin{enumerate}[1.]
\item Write $(1,2,3)$ as a linear combination of the vectors $(1,1,1)$, $(3,5,6)$ and $(7,8,9)$ using orthonormalization. Provide rounded numerical calculations, rather than exact ones.
\begin{proof}
Let $u_1=(1,1,1)$, $u_2=(3,5,6)$ and $u_3=(7,8,9)$. Then orthogonalizing gives
$$v_1=0.58u_1=(0.58,0.58,0.58)$$
\begin{align*}v_2'&=u_2-(u_2,v_1)v_1\\&=u_2-8.08v_1=(-1.67,0.33,1.33)\end{align*}
$$v_2=v_2'/2.16=(-0.77,0.15,0.62)$$
\begin{align*}v_3'&=u_3-(u_3,v_1)v_1-(u_3,v_2)v_2\\&=u_3-13.86v_1-1.39v_2=(0.07,-0.21,0.14)\end{align*}
$$v_3=v_3'/0.27=(0.27,-0.80,0.53)$$
We can then write 
\begin{align*}(1,2,3)&=\langle (1,2,3), v_1\rangle v_1+\langle (1,2,3), v_2\rangle v_2+\langle (1,2,3), v_3\rangle v_3
\\&=3.46v_1+1.39v_2+0.27v_3
\end{align*}
Plugging in the formulas for $v_1, v_2, v_3$ in terms of $u_1, u_2, u_3$ we get:
\begin{align*}
(1,2,3)&=3.46(u_1/.58)+1.39(2.16(u_2-8.08v_1))+0.27(0.27(u_3-13.86v_1-1.39v_2))
\\&=5.67u_1+1.39(2.16u_2-17.45v_1)+0.7 u_3-1.01 v_1-0.10v_2
\end{align*}
\end{proof}
\item Let $\Norm{\cdot}$ be the norm in $\R^2$ for which the unit circle is a regular hexagon with side length 1. Prove that $\Norm{\cdot}$ is not induced by an inner product.
\item Let $f_i(x)=x^i$, $i\in \N_{\geq0}$, be the basis of monomials in $C_2[a,b]$.
\begin{enumerate}
\item Is it true that every continuous function on $C_2[a,b]$ is of the form $\sum_{i=0}^\infty a_if_i$? (Hint: take a derivative.)
\item Let $g_i$ be the associated orthonormal basis.  Is it true that every continuous function on $C_2[a,b]$ is of the form $\sum_{i=0}^\infty a_ig_i$? 
\item Why don't the two results contradict each other?
\end{enumerate}
\end{enumerate}





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